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<h1 class="title"><img src="entitle.gif" width="276" height="113" alt="ENT" /></h1>
<h2 class="title">
A Pseudorandom Number Sequence Test Program
</h2>

<p />

<hr />

<p>
This page describes a program, <b>ent</b>, which applies various
tests to sequences of bytes stored in files and reports the results of
those tests.  The program is useful for evaluating pseudorandom
number generators for encryption and statistical sampling
applications, compression algorithms, and other applications where the
information density of a file is of interest.
</p>


<h3>NAME</h3>
     <b>ent</b> - pseudorandom number sequence test

<h3>SYNOPSIS</h3>

     <b>ent</b> [ <b>-b -c -f -t -u</b> ] [ <i>infile</i> ]

<h3>DESCRIPTION</h3>

<p>
<b>ent</b> performs a variety of tests on the stream of
bytes in <i>infile</i> (or standard input if no <i>infile</i>
is specified) and produces output as follows on the standard
output stream:
</p>

<pre>
    Entropy = 7.980627 bits per character.

    Optimum compression would reduce the size
    of this 51768 character file by 0 percent.
 
    Chi square distribution for 51768 samples is 1542.26, and randomly
    would exceed this value less than 0.01 percent of the times.
  
    Arithmetic mean value of data bytes is 125.93 (127.5 = random).
    Monte Carlo value for Pi is 3.169834647 (error 0.90 percent).
    Serial correlation coefficient is 0.004249 (totally uncorrelated = 0.0).
</pre>

<p>
The values calculated are as follows:
</p>

<dl class="options">
<dt><b>Entropy</b></dt>
<dd>The information density of the contents of the file,
                expressed as a number of bits per character.  The
                results above, which resulted from processing an image
                file compressed with JPEG, indicate that the file is
                extremely dense in information&mdash;essentially random.
                Hence, compression of the file is unlikely to reduce
                its size.  By contrast, the C source code of the
                program has entropy of about 4.9 bits per character,
                indicating that optimal compression of the file would
                reduce its size by 38%.  [Hamming, pp.&nbsp;104&ndash;108]</dd>


<dt><b>Chi-square Test</b></dt>
<dd><p>The chi-square test is the most commonly used test for
                the randomness of data, and is extremely sensitive to
                errors in pseudorandom sequence generators.  The
                chi-square distribution is calculated for the stream
                of bytes in the file and expressed as an absolute
                number and a percentage which indicates how frequently
                a truly random sequence would exceed the value
                calculated.  We interpret the percentage as the degree
                to which the sequence tested is suspected of being
                non-random.  If the percentage is greater than 99% or
                less than 1%, the sequence is almost certainly not
                random.  If the percentage is between 99% and 95% or
                between 1% and 5%, the sequence is suspect.
                Percentages between 90% and 95% and 5% and 10%
                indicate the sequence is &ldquo;almost suspect&rdquo;.  Note that
                our JPEG file, while very dense in information, is far
                from random as revealed by the chi-square test.
</p>

<p>
                Applying this test to the output of various
                pseudorandom sequence generators is interesting.  The
                low-order 8 bits returned by the standard Unix
                <code>rand()</code> function, for example, yields:
</p>

<blockquote>
                  Chi square distribution for 500000 samples is 0.01,
                  and randomly would exceed this value more than 99.99 percent
                  of the times.
</blockquote>

<p>
                While an improved generator [Park &amp; Miller] reports:
</p>

<blockquote>
                  Chi square distribution for 500000 samples is
                  212.53, and randomly would exceed this value 97.53
                  percent of the times.
</blockquote>

<p>
                Thus, the standard Unix generator (or at least the
                low-order bytes it returns) is unacceptably
                non-random, while the improved generator is much
                better but still sufficiently non-random to cause
                concern for demanding applications.  Contrast both of
                these software generators with the chi-square result
                of a genuine random sequence created by
		<a href="http://www.fourmilab.ch/hotbits/">timing
                radioactive decay events</a>.
</p>

<blockquote>
    	    	  Chi square distribution for 500000 samples is 249.51, and randomly
    	    	  would exceed this value 40.98 percent of the times.
</blockquote>

<p>
                See [Knuth, pp.&nbsp;35&ndash;40] for more information on the
                chi-square test.  An interactive
                <a href="http://www.fourmilab.ch/rpkp/experiments/analysis/chiCalc.html">chi-square
                calculator</a> is available at this site.
</p>
</dd>

<dt><b>Arithmetic Mean</b></dt>
<dd>This is simply the result of summing the all the bytes
                (bits if the <b>-b</b> option is specified) in the file
                and dividing by the file length.  If the data are
                close to random, this should be about 127.5 (0.5
                for <b>-b</b> option output).  If the
                mean departs from this value, the values are
                consistently high or low.</dd>

<dt><b>Monte Carlo Value for Pi</b></dt>
<dd>Each successive sequence of six bytes is used as 24 bit X and Y
                co-ordinates within a square.  If the distance of the
                randomly-generated point is less than the radius of a
                circle inscribed within the square, the six-byte
                sequence is considered a &ldquo;hit&rdquo;.  The percentage of
                hits can be used to calculate the value of Pi.  For
                very large streams (this approximation converges very
                slowly), the value will approach the correct value of
                Pi if the sequence is close to random.  A 500000 byte
                file created by radioactive decay yielded:

<blockquote>
                  Monte Carlo value for Pi is 3.143580574 (error 0.06 percent).
</blockquote>
</dd>

<dt><b>Serial Correlation Coefficient</b></dt>
<dd>This quantity measures the extent to which each byte in the file
                depends upon the previous byte.  For random sequences,
                this value (which can be positive or negative) will,
                of course, be close to zero.  A non-random byte stream
                such as a C program will yield a serial correlation
                coefficient on the order of 0.5.  Wildly predictable
                data such as uncompressed bitmaps will exhibit serial
                correlation coefficients approaching 1.  See [Knuth,
                pp.&nbsp;64&ndash;65] for more details.</dd>
</dl>

<h3>OPTIONS</h3>

<dl class="options">
<dt><b>-b</b></dt>  <dd>The input is treated as a stream of bits rather
                than of 8-bit bytes.  Statistics reported reflect
                the properties of the bitstream.</dd>

<dt><b>-c</b></dt>  <dd>Print a table of the number of occurrences of
               each possible byte (or bit, if the <b>-b</b> option
               is also specified) value, and the fraction of the
               overall file made up by that value.  Printable
               characters in the ISO 8859-1 Latin-1 character set
               are shown along with their decimal byte values.
               In non-terse output mode, values with zero
               occurrences are not printed.</dd>

<dt><b>-f</b></dt>  <dd>Fold upper case letters to lower case before
               computing statistics.  Folding is done based on the
               ISO 8859-1 Latin-1 character set, with accented
               letters correctly processed.</dd>

<dt><b>-t</b></dt>  <dd>Terse mode: output is written in Comma
               Separated Value (CSV) format, suitable for
               loading into a spreadsheet and easily read
               by any programming language.  See
               <a href="#Terse">Terse Mode Output Format</a>
               below for additional details.</dd>

<dt><b>-u</b></dt>  <dd>Print how-to-call information.</dd>
</dl>

<h3>FILES</h3>

<p>
    If no <i>infile</i> is specified, <b>ent</b> obtains its input
    from standard input.  Output is always written to standard
    output.
</p>

<h3><a name="Terse" class="i">TERSE MODE OUTPUT FORMAT</a></h3>

<p>
Terse mode is selected by specifying the <b>-t</b> option
on the command line.  Terse mode output is written in
Comma Separated Value (CSV) format, which can be directly
loaded into most spreadsheet programs and is easily read
by any programming language.  Each record in the CSV
file begins with a record type field, which identifies
the content of the following fields.  If the <b>-c</b>
option is not specified, the terse mode output will
consist of two records, as follows:
</p>

<pre>
0,File-bytes,Entropy,Chi-square,Mean,Monte-Carlo-Pi,Serial-Correlation
1,<em>file_length</em>,<em>entropy</em>,<em>chi_square</em>,<em>mean</em>,<em>Pi_value</em>,<em>correlation</em>
</pre>

<p>
where the italicised values in the type 1 record are the
numerical values for the quantities named in the type 0
column title record.  If the <b>-b</b> option is specified, the second
field of the type 0 record will be &ldquo;<tt>File-bits</tt>&rdquo;, and
the <em>file_length</em> field in type 1 record will be given
in bits instead of bytes.  If the <b>-c</b> option is specified,
additional records are appended to the terse mode output which
contain the character counts:
</p>

<pre>
2,Value,Occurrences,Fraction
3,<em>v</em>,<em>count</em>,<em>fraction</em>
. . .
</pre>

<p>
If the <b>-b</b> option is specified, only two type 3 records will
appear for the two bit values <em>v</em>=0 and <em>v</em>=1.
Otherwise, 256 type 3 records are included, one for each
possible byte value.  The second field of a type 3 record
indicates how many bytes (or bits) of value <em>v</em>
appear in the input, and <em>fraction</em> gives the decimal
fraction of the file which has value <em>v</em> (which is
equal to the <em>count</em> value of this record divided by
the <em>file_length</em> field in the type 1 record).
</p>

<h3>BUGS</h3>

<p>
    Note that the &ldquo;optimal compression&rdquo; shown for the file is
    computed from the byte- or bit-stream entropy and thus
    reflects compressibility based on a reading frame of
    the chosen width (8-bit bytes or individual bits if the
    <b>-b</b> option is specified).  Algorithms which use
    a larger reading frame, such as the Lempel-Ziv [Lempel&nbsp;&amp;&nbsp;Ziv]
    algorithm, may achieve greater compression if the file
    contains repeated sequences of multiple bytes.
</p>

<h2><a href="http://www.fourmilab.ch/random/random.zip"><img
    src="http://www.fourmilab.ch/images/icons/file.gif"
    class="border0" alt="" align="top" width="40"
    height="40" /></a> <a href="http://www.fourmilab.ch/random/random.zip">Download
    	random.zip</a> (Zipped archive)</h2>

<p>
    The program is provided as <a href="http://www.fourmilab.ch/random/random.zip">random.zip</a>,
    a <a href="http://www.info-zip.org/">Zipped</a> archive
    containing an ready-to-run Win32 command-line program,
    <code>ent.exe</code> (compiled using Microsoft Visual C++ .NET,
    creating a 32-bit Windows executable), and in source code form
    along with a <code>Makefile</code> to build the program under
    Unix.
</p>

<h3>SEE ALSO</h3>

<dl class="options">
    	<dd><cite><a href="http://www.fourmilab.ch/rpkp/experiments/statistics.html">Introduction
              to Probability and Statistics</a></cite></dd>

<dt>[Hamming]</dt> <dd>Hamming, Richard W.
              <cite>Coding and Information Theory</cite>.
              Englewood Cliffs NJ: Prentice-Hall, 1980.</dd>

<dt>[Knuth]</dt>   <dd>Knuth, Donald E.
              <cite>
              <a href="http://www.amazon.com/exec/obidos/ASIN/0201896842/fourmilabwwwfour" target="Amazon_Fourmilab">
              The Art of Computer Programming,
              Volume 2 / Seminumerical Algorithms</a></cite>.
              Reading MA: Addison-Wesley, 1969.
              ISBN 0-201-89684-2.</dd>

<dt>[Lempel&nbsp;&amp;&nbsp;Ziv]</dt>
                <dd>Ziv J. and A. Lempel.
                &ldquo;A Universal Algorithm for Sequential Data Compression&rdquo;.
                <cite>IEEE Transactions on Information Theory</cite>
                <b>23</b>, 3, pp.&nbsp;337-343.</dd>

<dt>[Park &amp; Miller]</dt> <dd>Park, Stephen K. and Keith W. Miller.
                &ldquo;Random Number Generators: Good Ones Are Hard to Find&rdquo;.
                <cite>Communications of the ACM</cite>, October 1988, p.&nbsp;1192.</dd>
</dl>

<p />
<hr />

<p />
<blockquote class="rights">
     This software is in the public domain.  Permission to use, copy,
     modify, and distribute this software and its documentation for
     any purpose and without fee is hereby granted, without any
     conditions or restrictions.  This software is provided &ldquo;as is&rdquo;
     without express or implied warranty.
</blockquote>

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<address>by <a href="http://www.fourmilab.ch/">John Walker</a></address>
January 28th, 2008
</td>
<td class="right">
</td>
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<h3><a href="http://www.fourmilab.ch/">Fourmilab Home Page</a></h3>

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